We also demonstrated that the Egorov projective structure is variational in any dimension \(n+1\ge 3\) 
by exhibiting a Kropina type pseudo-Finsler metric. For \(n=2\)  this could be obtained from the
results of Douglas [1]}; in [2]} another Lagrangian was derived for
ODE (REF ) though without any relation to the Egorov structure.
By our previous work [3]} it is not metrizable
in the pseudi-Riemannian setting. In this work we proved it is not metrizable in the
Finsler setting. We also demonstrated variationability of some other notable path geometries with
many infinitesimal symmetries.
